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I added some simpler motivation in terms of the basic example to the beginning of distributive law.
Yes, nice. Rings make an obvious choice to illustrate a distributive law between monads. Then we’re introduced to the scary thought that there’s a whole 2-category of such laws. Could this be illustrated by a surprising example or two of a distributive law, perhaps something one would never have thought to be such a law?
one would never have thought to be such a law
The level of difficulty in fullfilling this task depends on who is "one" who never thought...
There’s definitely a gap between the first two sections of this page which someone who has some time on their hands should fill. (-:
a surprising example
I for one was quite tickled to find out that factorization systems are distributive laws between categories taken as monads in Span(Set). That might be a good non-obvious example.
Good. I’ve put that example in.
It’s not quite true as stated, though – ordinary distributive laws in Span(Set) only give you “strict” factorization systems. I corrected it.
I keep pointing to “distributive law” whenever I mention something distributive. Now looking back at the entry to see what it actually says, I notice that many of the common “distributivities” were not really mentioned. I added a quick pointer to distributive category and to tensor products distributing over direct sums. But this needs to be better incorporated into the entry, eventually.
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